Hmm. So the battery's a factor then.

I'm not specifically familiar with your specific hardware, but I'd try charging the laptop when it's off.

On PC laptops, my understanding is that normally, just the BIOS controls charging in that case, so that takes the OS out of the equation. If it's not charging in that situation, then I doubt that the operating system is going to be able to change the situation; the battery management system (BMS) on your hardware doesn't like the battery, for whatever reason. Dead, not actually the right model...shrugs Dunno.

From a Linux standpoint, you can pull whatever information you can about the battery looking in the virtual files under /sys/class/power_supply/BAT0. If there's no BAT entry, I'd guess that Linux can't even see a battery, that the BMS is saying that there's nothing there.

EDIT: I was thinking that maybe if there was a ribbon cable connecting the battery to the rest of the system, maybe it wasn't properly reconnected or something, but doing a really quick skim of this video replacing said battery:

https://www.youtube.com/watch?v=DONdmtR6mug

It looks like there is a ribbon cable, but it runs to a board containing the trackpad as well as the battery, and given that you haven't said anything about the trackpad not working, I assume that it's back in place.

EDIT2: In that video, it looks like the old battery is glued to the battery board, and the guy has to use some "adhesive softener" to get it out. If you did the same, I assume that there is liable to be some goop floating around in there. I assume that there are some contacts between the battery connector and the battery board; mayhap it or something like that got on top of the battery contacts. You might consider trying to clean those if other things don't work and the system can't see the battery.

EDIT3: Yeah, the top comment on that video is indeed someone who apparently didn't get the ribbon cable all the way back in, and their keyboard wasn't working.

The first time I replaced the back cover and tried logging in I couldn't type in my password. I followed Apple's hints but they didn't work so I took the cover back off again and refitted the flex cable. This time there was a satisfying little snap sound when I plugged it in at the logic board end and after that everything worked fine. Best to just put a couple of the screws back in the back cover and try logging in to make sure.

If you're not seeing that, I assume that the ribbon cable is properly connected.

EDIT4: Oh, the batttery has a second cable that goes right to the battery. The other cable just had to come off because it obstructs removing the battery. Sorry, comes from just doing a quick skim. Might check that that cable is in all the way.